Blog Post 8, WS 3.1, Problem 4: Missing Matter

We actually observe a flat rotation curve in our own Milky Way. (You will show this with a radio telescope in your second lab!) This means v(r) is nearly constant for a large range of distances.

(a) Lets call this constant rotational velocity Vc. If the mass distribution of the Milky Way is spherically symmetric, what must be the M(<r) as a function of r in this case, in terms of Vc, r, and G?

(b) How does this compare with the picture of the galaxy you drew last week with most of the mass appearing to be in bulge?

(c) If the Milky Way rotation curve is observed to be flat (Vc = 240 km/s) out to 100 kpc, what is the total mass enclosed within 100 kpc? How does this compare with the mass in stars? Recall the total mass of stars in the Milky Way, a number you have been given in your first assignment and should commit to memory.

(a) Lets call this constant rotational velocity Vc. If the mass distribution of the Milky Way is spherically symmetric, what must be the M(<r) as a function of r in this case, in terms of Vc, r, and G? 

Let's start with our answer for last time.  Now V is a constant, and we are solving for M.

\[V(r)=\left(\frac{GM_{enc}}{r}\right)^{1/2}\]
\[\boxed{M(<r)=\frac{V_C^2 r}{G}}\]
This implies that mass must increase as a function of \(r\).

(b) How does this compare with the picture of the galaxy you drew last week with most of the mass appearing to be in bulge? 

This is a disastrous contradiction.  Our new equation implies that the mass density must increase as a function of distance form the center of the galaxy, but a galaxy with the majority of the mass in the center would never exhibit this behavior.  Beyond the central bulge, the mass would increase very slowly, less and less as radius increased.  Something is wrong with our model.

(c) If the Milky Way rotation curve is observed to be flat (Vc = 240 km/s) out to 100 kpc, what is the total mass enclosed within 100 kpc? How does this compare with the mass in stars? Recall the total mass of stars in the Milky Way, a number you have been given in your first assignment and should commit to memory. 

Let's plug in numbers:

\[M(<r)=\frac{V_C^2 r}{G}=\frac{(2.4\times 10^7 cm/s)^2 \left(100kpc \times \frac{3.1\times 10^{21}cm}{1 kpc}\right)}{6.7\times 10^{-8}cm^3g^{-1}s^{-2}}=\boxed{2.7\times 10^{45}g}\]
The stellar mass of the Milky Way is \(5\times 10^{10} M_{\odot}\) which is about \(1\times 10^{44} g\).  The ratio of these two masses is 3.7%.  Thus, stars make up only 3.7% of the Milky Way's mass.  The resolution to this problem is the speculated existence of Dark Matter, matter that we cannot yet detect that leads to the gravitational effects that we measure on the galactic level.

I worked with B. Brzycki, G. Grell, and N. James on this problem.